Quantum Approximate Optimisation Applied to Graph Similarity
Nicholas J. Pritchard
TL;DR
This paper demonstrates a scalable quantum-classical approach to graph similarity by mapping maximal edge-overlap to the Quantum Approximate Optimisation Algorithm (QAOA) and implementing a compact permutation-based encoding. It introduces Qolab, a high-performance simulator capable of desktop-to-cluster scale experiments, and evaluates eight classical optimisers across six QAOA decompositions, highlighting the trade-offs between solution quality and computational cost. The work shows that a memory-efficient encoding, while increasing circuit complexity due to infeasible solutions, can be effectively simulated up to 22 qubits and provides detailed performance analyses, strengthening the case for using QAOA in challenging problems like graph similarity. Collectively, the study offers a practical framework and toolkit for validating QAOA on non-trivial problems, facilitating exploration of quantum advantages for near-term hardware and guiding future extensions to other combinatorial tasks.
Abstract
Quantum computing promises solutions to classically difficult and new-found problems through controlling the subtleties of quantum computing. The Quantum Approximate Optimisation Algorithm (QAOA) is a recently proposed quantum algorithm designed to tackle difficult combinatorial optimisation problems utilising both quantum and classical computation. The hybrid nature, generality and typically low gate-depth make it a strong candidate for near-term implementation in quantum computing. Finding the practical limits of the algorithm is currently an open problem. Until now, no tools to facilitate the design and validation of probabilistic quantum optimisation algorithms such as the QAOA on a non-trivial scale exist. Graph similarity is a long standing classically difficult problem withstanding decades of research from academia and industry. Determining the maximal edge overlap between all possible node label permutations is an NP-Complete task and provides an apt measure of graph similarity. We introduce a novel quantum optimisation simulation package facilitating investigation of all constituent components of the QAOA from desktop to cluster scale using graph similarity as an example. Our simulation provides flexibility and performance. We investigate eight classical optimisation methods each at six levels of decomposition. Moreover an encoding for permutation based problems such as graph similarity through edge overlap to the QAOA allows for significant quantum memory savings at the cost of additional operations. This compromise extends into the classical portion of the algorithm as the inclusion of infeasible solutions creates a challenging cost-function landscape. We present performance analysis of our simulation and of the QAOA setting a precedent for investigating and validating numerous other difficult problems to the QAOA as we move towards realising practical quantum computation.